RESEARCH STATEMENT ADAM CHAPMAN

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1 RESEARCH STATEMENT ADAM CHAPMAN My research focuses mainly on central simple algebras and related algebraic objects, such as Clifford algebras and quadratic forms. My research interest encompasses also ring theory and computational algebra. 1. Background Simple algebras (or rings) are the building blocks of ring theory. We focus on simple algebras of finite dimension over their center (a field), i.e. central simple algebras. Apart from their theoretical significance, such algebras have long been proven to be useful in Physics (e.g. in Dirac s treatment of the relativistic wave equation), and nowadays they are used in error-correcting codes as well. Given a field F, the Brauer equivalence classes of central simple F - algebras form a torsion group Br(F ). Given a positive integer d and assuming char(f ) d or = 0 and F contains a primitive dth root of unity, dbr(f ) is known to be generated (by the renown Merkurjev-Suslin theorem proved in [MS82]) by algebras of the form (α, β) d,f = F x, y : x d = α, y d = β, yx = ρxy for some α, β F. Such algebras are called symbol algebras. An equivalent theorem had been proven earlier by Albert in case d = char(f ) (see [Alb61]). In this case, the generators of d Br(F ) are the algebras of the form [α, β) p,f = F x, y : x p x = α, y p = β, yx xy = y for some α F and β F. Considering the symbol algebras as letters and the tensor products as words, there are two very natural questions: Question 1.1 ( The Word Problem ). Given two words, do the represent they same element in the group or not? Question 1.2 ( The Symbol Length Problem ). How many letters does one need in order to spell a given word? 1

2 2 ADAM CHAPMAN The first question is the motivation behind the chain lemma (see Section 2) and the second question is one of the motivations behind the study of Kummer spaces (see Section 3). 2. Chain Lemma As part of the attempt of solving the word problem, we want to understand the situation when two words are equivalent. The chain lemma is a theorem which describes the equivalence between two givetn words by giving a sequence of equivalent words where every two consecutive words share most of the letters. Chain lemmas for tensor products of more than one symbol algebra were provided only for quaternion algebras. The following theorem sums up the state of the arts: Theorem 2.1. ([Siv12], [Cha12, Theorem 5.2] and [Cha15a, Theorem 0.1]) Assuming char(f ) 2 and either n = 2 or I 3 F = 0, every two isomorphic tensor products of n quaternion algebras are connected by a chain of steps that change up to two quaternion algebras at a time. Instead of looking at tensor products of symbol algebras, one can take an algebra that decomposes in such a way and look at set of generators (x 1, y 1 x 2, y 2... x n, y n ) such that every pair x i, y i satisfies x 2 i, y2 i F and y i x i = x i y i, and generators from different pairs commute. Such a set of generators gives rise to a decomposition of the algebra as a tensor product of quaternion algebras. The chain lemma on tensor products of symbol can be deduced from a chain lemma for sets of generators, assuming the latter is proved. Such a chain is in general more difficult to produce. For biquaternion algebras we have such a chain lemma: Theorem 2.2. ([CV13, Theorem 4.3]) Every two sets of generators of a biquaternion algebra are connected by a chain of steps that change up to two generators each time. In special cases we can also say something about tensor products of more than 2 quaternion algebras: Theorem 2.3. ([BC15, Theorem 4.5]) Assuming cd 2 (F ) = 2 and F is 2-special, every two sets of generators of a tensor product of quaternion algebras are connected by a steps preserving at least one generator at a time. It is yet unclear what happens when we drop the assumption cd 2 (F ) = 2.

3 RESEARCH STATEMENT 3 3. Kummer spaces and Clifford algebras Given a central simple F -algebra A of exponent d, a Kummer space in A is an F -vector space V satisfying v d F for any v V. Let ϕ : V F be the exponentiation form of V, i.e. ϕ(v) = v d for any v V. This is a homogeneous polynomial form. One can associate to ϕ a Clifford algebra C ϕ, i.e. the quotient of the tensor algebra of V by the relations ϕ(v) = v d for all v V. Considering the natural embedding of A in a matrix algebra, it becomes a representation of C ϕ. There are several natural questions that arise in this context: Question 3.1. Given a central simple algebra, what is the maximal dimension of a Kummer subspace? Question 3.2. Given a homogeneous polynomial form ϕ, what are the minimal and maximal irreducible representations of C ϕ? A solution to Question 3.1 will provide a lower bound for the dimension of an irreducible representation of Clifford algebras - a partial solution to Question 3.2. Question 3.1 is also connected to the symbol length question, for Kummer spaces in tensor products of symbol algebras have been recently used by Matzri to give an upper bound for the symbol length of central simple algebras over C r -fields, and their actual dimension appeared in the formula. We conjecture that dm + 1 is the maximal dimension of Kummer spaces in division tensor products of m symbol algebras of degree d. This is obvious when d = 2. For d = 3 and n = 1 this is also known. In a recent collaboration with Grynkiewicz, Matzri, Rowen, Vishne and myself, we proved that the conjecture holds for the generic symbol algebra of prime degree. I also proved the conjecture in the generic case for d = 3 and any number n. In another collaborative project with Ure and myself, we proved the conjecture in the generic case for d = 4 and any n. The structure of Clifford algebras is usually very complicated, unless we consider Clifford algebras of quadratic forms. These algebras have therefore been studied via certain quotients. Theorem 3.3. ([CV12]) If ϕ is a binary form of prime degree p then C ϕ has an Azumaya quotient whose center is the function field of a hyper-elliptic curve and its simple quotients are symbol algebras of degree p. This theorem generalized the main result of [Hai84] which studied the cubic case, where the quotient is C ϕ itself. Representations of the Clifford algebras have been studied via their correspondence with Ulrich bundles. In [VdB87] this correspondence was pointed

4 4 ADAM CHAPMAN out and it was proved that if ϕ is binary and of degree 4 then C ϕ has irreducible representations of unbounded high ranks. A similar result was obtained for ternary cubic forms in [CKM12]. The notion of the Clifford algebra was generalized in several different ways. One was done by Pappacena in [Pap00]: Given a homogeneous polynomial φ(w, u 1,..., u n ) = w d + f 1 (u 1,..., u n )w d f d (u 1,..., u n ) which is monic with respect to the first variable, we define C φ to be F x 1,..., x n : x d +f 1 (u 1,..., u n ) x d 1 + +f d (u 1,..., u n ) = 0 u 1,..., u n F where x = u 1 x u n x n. This extended form of the Clifford algebra captures all central simple algebras as simple quotients by taking the characteristic polynomial of each algebra. This algebra was studied in [CK15] where the known results on the irreducible representations of the standard Clifford algebra in case d = 3 and n = 2, or d 4 and n = 2 or d = n = 3 were obtained also for this generalized algebra. Together with Krashen and Lieblich, we have recently studied the Clifford algebra associated to a morphism of proper varieties, which generalizes the Clifford-Pappacena construction and other constructions. 4. Other directions There are other ways of studying central simple algebras. When applications are concerned, the computational aspects of the theory, such as computing polynomial equations over central simple algebras and computing eigenvalues of matrices over such algebras, gain in importance. I explored these topics in [Cha12] and [Cha15b], and still have ongoing projects on related questions. When structure theory is concerned, one aspect that has gained popularity recently is the study of subfields of central simple algebras, when algebras share a subfield and when algebras share all their subfields (in which case we say they have the same genus ). In finite characteristic there is a difference between inseparable and separable subfields, and sharing an inseparable one seems to be a stronger property than sharing a separable one. This was known for quaternion algebras and I have recently generalized it to cyclic p-algebras of prime degree. In a collaborative project with Dolphin and Laghribi, we studied the genus of quaternion algebras over fields of characteristic 2.

5 RESEARCH STATEMENT 5 References [Alb61] A. Adrian Albert, Structure of algebras, Revised printing. American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., MR (23 #A912) [BC15] Demba Barry and Adam Chapman, Square-central and Artin Schreier elements in division algebras, Arch. Math. (Basel) 104 (2015), no. 6, MR [Cha12] Adam Chapman, General polynomials over division algebras and left eigenvalues, Electron. J. Linear Algebra 23 (2012), MR [Cha15a], Addendum: Chain equivalences for symplectic bases, quadratic forms and tensor products of quaternion algebras, J. Algebra Appl. 14 (2015), no. 7, (2 pages). MR [Cha15b], Quaternion quadratic equations in characteristic 2, J. Algebra Appl. 14 (2015), no. 3, , 8. MR [CK15] Adam Chapman and Jung-Miao Kuo, On the generalized Clifford algebra of a monic polynomial, Linear Algebra Appl. 471 (2015), MR [CKM12] Emre Coskun, Rajesh S. Kulkarni, and Yusuf Mustopa, On representations of Clifford algebras of ternary cubic forms, New trends in noncommutative algebra, Contemp. Math., vol. 562, Amer. Math. Soc., Providence, RI, 2012, pp MR [CV12] Adam Chapman and Uzi Vishne, Clifford algebras of binary homogeneous forms, J. Algebra 366 (2012), MR [CV13], Square-central elements and standard generators for biquaternion algebras, Israel J. Math. 197 (2013), no. 1, MR [Hai84] Darrell E. Haile, On the Clifford algebra of a binary cubic form, Amer. J. Math. 106 (1984), no. 6, MR (86c:11028) [MS82] A. S. Merkur ev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, , MR (84i:12007) [Pap00] Christopher J. Pappacena, Matrix pencils and a generalized Clifford algebra, Linear Algebra Appl. 313 (2000), no. 1-3, MR (2001e:15010) [Siv12] A. S. Sivatski, The chain lemma for biquaternion algebras, J. Algebra 350 (2012), MR (2012j:16037) [VdB87] M. Van den Bergh, Linearisations of binary and ternary forms, J. Algebra 109 (1987), no. 1, MR (88j:11020)